Question: Determine how many solutions exist for the system of equations. ${5x-y = -2}$ ${15x-3y = 27}$
Answer: Convert both equations to slope-intercept form: ${5x-y = -2}$ $5x{-5x} - y = -2{-5x}$ $-y = -2-5x$ $y = 2+5x$ ${y = 5x+2}$ ${15x-3y = 27}$ $15x{-15x} - 3y = 27{-15x}$ $-3y = 27-15x$ $y = -9+5x$ ${y = 5x-9}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 5x+2}$ ${y = 5x-9}$ Both equations have the same slope with different y-intercepts. This means the equations are parallel. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Parallel lines never intersect, thus there are NO SOLUTIONS.